Recently, the demand for faster and more reliable data transmission has brought up complex communications systems. As a result, it has become more difficult to carry out closed-form solutions that can provide insight about performance levels. In this paper, different from the existing research, we study a cognitive radio system that employs hybrid-automatic-repeat-request (HARQ) protocols under quality-of-service (QoS) constraints. We assume that the secondary users access the spectrum by utilizing a strategy that is a combination of underlay and interweave access techniques. Considering that the secondary users imperfectly perform channel sensing in order to detect the active primary users and that there is a transmission deadline for each data packet at the secondary transmitter buffer, we formulate the state-transition model of the system. Then, we obtain the state-transition probabilities when HARQ-chase combining is adopted. Subsequently, we provide the packet-loss rate in the channel and achieve the effective capacity. Finally, we substantiate our analytical derivations with numerical results.